MECHANICAL PROCESSES IN...

MECHANICAL PROCESSES IN BIOCHEMISTRY

Carlos Bustamante,1–3 Yann R. Chemla,3

Nancy R. Forde,1 and David Izhaky1

1Howard Hughes Medical Institute and the Departments of 2Molecular and CellBiology, and 3Physics, University of California, Berkeley, California 94720-3206;email: [email protected], [email protected],[email protected], [email protected]

Key Words mechanical forces, single-molecule manipulation, molecularmotors, mechanical unfolding, enzyme catalysis

f Abstract Mechanical processes are involved in nearly every facet of the cellcycle. Mechanical forces are generated in the cell during processes as diverse aschromosomal segregation, replication, transcription, translation, translocation ofproteins across membranes, cell locomotion, and catalyzed protein and nucleic acidfolding and unfolding, among others. Because force is a product of all these reactions,biochemists are beginning to directly apply external forces to these processes to alterthe extent or even the fate of these reactions hoping to reveal their underlyingmolecular mechanisms. This review provides the conceptual framework to under-stand the role of mechanical force in biochemistry.

CONTENTS

PROLOGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706THE EFFECT OF FORCE ON THE THERMODYNAMICS AND KINETICS OF

CHEMICAL REACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707Effect of Force on the Free Energy of a Reaction . . . . . . . . . . . . . . . . . . . 709Effect of Force on the Kinetics of a Reaction . . . . . . . . . . . . . . . . . . . . . 713

MECHANICAL UNFOLDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715Irreversibility in Mechanical Unfolding Experiments . . . . . . . . . . . . . . . . . 716The Unfolding Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717Relating Mechanical Stability to Local Molecular Structure . . . . . . . . . . . . . 722Extending Single-Molecule Mechanical Properties to the Cellular Level . . . . . 724

MOLECULAR MOTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725Mechanical Properties of Molecular Motors . . . . . . . . . . . . . . . . . . . . . . 726Mechanochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

STRAIN IN ENZYME CATALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

Annu. Rev. Biochem. 2004. 73:705–48doi: 10.1146/annurev.biochem.72.121801.161542

Copyright © 2004 by Annual Reviews. All rights reserved

7050066-4154/04/0707-0705$14.00

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The Catalytic Advantage of Transition State Complementarity . . . . . . . . . . . 737Experimental Evidence for Strain-Induced Catalysis . . . . . . . . . . . . . . . . . 740The Magnitude of the Enzyme-Substrate Forces . . . . . . . . . . . . . . . . . . . . 741Enzymes as Mechanical Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743

EPILOGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

PROLOGUE

Fifty years ago, biochemists described cells as small vessels that contain acomplex mixture of chemical species undergoing reactions through diffusion andrandom collision. This description was satisfactory inasmuch as the intricatepathways of metabolism and, later, the basic mechanisms of gene regulation andsignal transduction were still being unraveled. Gradually, and in part as a resultof the parallel growth in our structural understanding of the molecular compo-nents of the cell, the limitations of this “chemical reactor” view of the cellbecame plain. Armed with a more precise knowledge of the structural bases ofmolecular interactions, the focus shifted more and more to the mechanisms bywhich these molecular components recognize and react with each other. More-over, it also became clear that cells are polar structures and that the cell interioris neither isotropic nor homogeneous; that many of the essential processes of thecell, such as chromosomal segregation, translocation of organelles from one partof the cell to another, protein import into organelles, or the maintenance of avoltage across the membrane, all involve directional movement and transport ofchemical species, in some cases against electrochemical gradients. Processessuch as replication, transcription, and translation require directional readout ofthe information encoded in the sequence of linear polymers. Slowly, the oldparadigm was replaced by one of a small “factory” of complex molecularstructures that behave in machine-like fashion to carry out highly specialized andcoordinated processes. These molecular machines are often complex assembliesof many proteins and contain parts with specialized functions, for example, asenergy transducers or molecular motors, converting chemical energy (either in theform of binding energy, chemical bond hydrolysis, or electrochemical gradients) intomechanical work through conformational changes and displacements.

To understand the behavior of this molecular machinery requires a fundamen-tal change in our conceptual and practical approaches to biochemical research.The cell, it appears, resembles more a small clockwork device than a reactionvessel of soluble components. Many of the functions of this device (whichbesides replication, transcription, translation, and organelle transport, include cellcrawling, cell adhesion, protein folding, protein and nucleic acid unfolding,protein degradation, and protein and nucleic acid splicing) are indeed mechanicalprocesses, and basic physical concepts such as force, torque, work, energyconversion efficiency, mechanical advantage, etc., are needed to describe them.The recent development of experimental methods that permit the direct

706 BUSTAMANTE ET AL.

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mechanical manipulation of single molecules now allows many of these mechan-ical processes to be investigated directly and in real-time fashion. Analyses of thedata so obtained also require the reformulation of many of the traditionalconcepts of thermodynamics and kinetics to incorporate terms corresponding toforces and torques.

This article attempts to critically review the most recent conceptual andexperimental developments in the mechanical characterization of biochemicalprocesses. In the following section we reformulate some of the main results ofthermodynamics and kinetics in terms of the effect of mechanical force toprovide a conceptual framework for the interpretation of the results presentedlater in this review. In the third section, we review the use of mechanical forceto unfold proteins and nucleic acids. Here, as in the following sections, wedescribe and illustrate the important new information that can be derived fromthe mechanical characterization of molecules and molecular processes rather thanproviding an exhaustive guide to the literature. In the fourth section we describethe mechanical properties of molecular motors, illustrate how mechanical forceis used to investigate their mechanisms of mechanochemical transduction, anddiscuss the many cellular functions now known to be mechanical processes. Thefinal section presents our current understanding of the importance of force andstrain in enzyme catalysis: how an otherwise silent form of chemical energy (thatassociated with binding interactions) can be and is used by enzymes to acceleratethe rate of chemical reactions in the cell.

The mechanical characterization of the cellular “factory” is just beginning.Many more mechanical cellular functions are likely to be discovered in thefuture. This exciting new aspect of the inner workings of the cell challenges usto learn to think in terms of concepts heretofore alien to the trained biochemist.We hope that this review will be a helping step in that direction.

THE EFFECT OF FORCE ON THE THERMODYNAMICSAND KINETICS OF CHEMICAL REACTIONS

Introduction

Many biochemical reactions proceed via large conformational changes within orbetween interacting molecules. Such conformational changes, which mayinvolve a combination of linear and rotational degrees of freedom, provideconvenient, well-defined mechanical reaction coordinates that can be used tofollow the progress of the reaction. Examples of these reaction coordinates arethe end-to-end distance of a molecule as it is being stretched, the position of amolecular motor as it moves along a track, the angle of a rotary motor’s shaft, orthe deformation of an enzyme’s binding pocket when it binds the substrate. Theeffect of an applied force can yield valuable information about the free energysurface of the reaction. In this section, we describe, using basic thermodynamic

707MECHANICAL PROCESSES IN BIOCHEMISTRY

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and kinetic relationships, how an applied mechanical force affects the freeenergy, equilibrium, and rate of a reaction occurring along a mechanical reactioncoordinate (for a more detailed treatment, see 1). For brevity, and because of itsdirect relevance to the majority of the examples in this review, we limit ourdiscussion to linear mechanical reaction coordinates. By replacing linear distanceby angle and force by torque, it is possible to derive analogous expressions forthe effect of torsion.

From thermodynamics, the energy change of a system (e.g., a molecule beingstretched, a motor moving along a track, etc.) can be separated into componentsrelated to the heat exchanged and the work done on or performed by the system.When the energy changes slowly enough that the system remains in quasi-staticequilibrium, these quantities are the reversible exchanged heat and the reversiblework (pressure-volume work and mechanical work):

dE � dqrev � dwrev

� (TdS) � ( � PdV � � F � dx).1.

In practice, the temperature and pressure are usually the independent variables inan experiment, and the Gibbs free energy (G � E � TS � PV) provides a morerelevant expression:

dG � � SdT � VdP � Fdx. 2.

At constant temperature and pressure, the work required to extend the system byan amount �x is

W � �x0

x0 � �x

Fdx. 3.

If the extension is carried out slowly enough that the system remains inquasi-static equilibrium, the work in Equation 3 is reversible and is equal to thefree energy change of the system, �Gstretch(�x). The work done to stretchthe molecule is positive, as both F and �x are positive. Positive work means thatthe surroundings have done work on the system; the free energy of the moleculeis increased.

Work is required to stretch a DNA molecule by its ends, and the amount offorce required to extend its ends a given distance is described by the worm-likechain model of polymer elasticity (2, 3). When the molecule is relaxed, thetension in the DNA molecule decreases and the molecule does work on itssurroundings. The molecule follows the same force-extension curve upon stretch-ing as relaxation, implying that these processes are occurring at equilibrium.Thus, DNA acts as a reversible spring, storing elastic energy as it is stretched andrestoring that same energy to the surrounding bath as it relaxes. In other words,


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mechanical stretching and relaxation of DNA are 100% efficient, when all ofinput (or stored) work is converted to a free energy change of the molecule andnone is dissipated as heat. By integrating the area under the force-extension curveof DNA, the free energy change in the molecule due to stretching [�Gstretch(x)]can thus be determined (Equation 3). Furthermore, by measuring the free energychange as a function of temperature, it is possible to determine the change inentropy, �Sstretch, and the change in enthalpy, �Hstretch, upon stretching themolecule. Recalling that the enthalpy H � G-TS and using Equation 2, it canbe shown that:

�Sstretch( x) � � (��Gstretch(x)

�T )P, x

, �Hstretch(x) � (�(�Gstretch(x)/T)

�(1/T) )P, x

. 4.

Effect of Force on the Free Energy of a Reaction

The effect of force on a reaction in which A is converted into B is illustrated bythe two-state system depicted in Figure 1. Here, A and B are distinct observablestates of the system; each occupies a local free energy minimum, at position xA

and xB, respectively, separated by a distance �x along the mechanical reactioncoordinate shown in Figure 1. A and B could represent the states of a motor insequential locations along its track, or folded and unfolded states of a protein. Atzero force, the free energy difference between A and B is simply

�G(F � 0) � �G0 � kBT ln[B]

[A], 5.

where �G0 is the standard state free energy, and [A] and [B] are the concentra-tions (more appropriately, the activities) of the molecule in states A and B. Sinceconcentrations are only well-defined in bulk measurements, [A] and [B] representprobabilities of populating states A and B in single-molecule experiments. To afirst approximation, an applied force “tilts” the free energy surface along themechanical reaction coordinate by an amount linearly dependent on distance(Equation 2), such that

�G(F) � �G0 � F(xB � xA) � kBT ln[B]

[A]. 6.

At equilibrium, �G � 0 and

�G0 � F�x � � kBT ln[B]eq(F)

[A]eq(F)� � kBT ln Keq(F).

7.

Thus, the equilibrium constant Keq(F) depends exponentially on the appliedforce. By applying a force assisting (F � 0) or opposing (F � 0) the transition,we can alter the equilibrium of the reaction, increasing or reducing the population


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Figure 1 The effect of force on the free energy of a two-state system, where xrepresents the mechanical reaction coordinate. (a) No applied force. (b) Solid curve:positive applied force. Dashed curve: no applied force. The application of forcelowers the energy of both the transition state ‡ and state B relative to state A (�G0‡

and �G0), which increases the rate of the forward reaction and the population of stateB, respectively. The positions of the free energy minima (xA and xB) and maximum(x‡) shift to longer and shorter x, respectively, with a positive applied force. Theirrelative shifts in position depend on the local curvature of the free energy surface. Thefree energy change of states A and B upon stretching is �Gstretch; see text.


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of state B relative to state A, respectively. Furthermore, from Equation 7, one candetermine the separation, �x, between states A and B from the slope of the plotof lnKeq(F) against the force.

For simplicity, we have assumed that the positions of states A and B, xA andxB, are unaffected by an applied force. In general, this assumption is not valid. Asan example of the reaction A3B, consider the mechanical unfolding of a proteinby its ends. As shown in Figure 1, force not only shifts the equilibrium towardstate B (the unfolded state), but also increases the average end-to-end distance xB

of the unfolded molecule from that of a zero-force, random-coil configurationxB(F � 0) to that of an extended polypeptide chain xB(F). It is clear from Figure1 that the shifts in xA and xB with force depend inversely on the local curvatureof the potential: the steeper the potential well, the more “localized” the state, andthe lesser the effect of force on its position. (For a harmonic potential well, theminimum shifts by F/G�, where G� is the second derivative of the free energysurface at the minimum.) In Figure 1, the end-to-end distance of the foldedprotein (xA) is less shifted by force than that of the unfolded molecule (xB).

Because the free energy of the reaction A3B under an applied force F mustbe measured between the new free energy minima at xB(F) and xA(F), Equation7 must be corrected to account for the small free energy change due to this shiftin minima:

�G0 � F�x � �GstretchA3 B (F) � � kBT ln Keq(F) 8.

where �x � xB(F � 0) � xA(F � 0) and �GstretchA3 B (F) is given by

�GstretchA3 B(F) � �Gstretch,B(F) � �Gstretch, A(F)

� �xB(F � 0)

xB(F)

FBdxB� �xA(F � 0)

xA(F)

FAdxA.9.

The two terms in Equation 9 are the free energy differences due to the shift in theminimum at state B [i.e., the free energy of stretching the molecule from xB(0)to xB(F)] and at state A [stretching from xA(0) to xA(F)], respectively (see Figure1). If states A and B have the same curvature, their minima are shifted by thesame amounts. In this situation, the two terms in Equation 9 cancel out exactly,and

�GstretchA3 B(F) � 0.

This would be the case for a molecular motor moving along a homogeneoustrack, where states A and B represent sequential positions on the track.

We have seen that force can shift the equilibrium of a reaction. In principle,it is possible to apply a force, F1/2, such that the equilibrium constant Keq � 1.At F1/2, the molecule has an equal probability of being in state B or A, and will


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thus spend half its time in each state. From Equation 8, it is evident that F1/2

provides a direct measure of the standard state free energy:

F1/ 2 � �x � �G0 � �GstretchA3 B (F), 10.

where here we include the contribution of the free energy of stretching, �Gstretch.Furthermore, by taking derivatives of Equation 10 with respect to temperatureand inverse temperature (as in Equation 4), one can determine the entropy andenthalpy, respectively, of the reaction:

�(F1/2 � �x)

�T� � �S0 � �Sstretch

A3 B (F),

�(F1/2 � �x/T)

�(1/T)� �H0 � �Hstretch

A3 B (F) .11.

The effect of force on the free energy of a reaction is illustrated experimentallyby the mechanical unfolding of a simple RNA hairpin (4, 5). The force-extensioncurve of the molecule (Figure 2a) shows a discontinuity at 15 pN, where thelength suddenly increases, due to the hairpin unfolding. Upon relaxation, themolecule follows the same force-extension curve, indicating that the unfoldingprocess is occurring reversibly. From the area under the transition, Equation 3can be used to calculate the free energy of unfolding the molecule mechanically,�G0 � �Gstretch. By correcting this value for �Gstretch, the standard free energy

Figure 2 Mechanical unfolding of the 22-bp P5ab RNA hairpin, a domain of thegroup I intron of T. thermophila (4). (a) The force-extension curve shows adiscontinuity at 15 pN, due to the hairpin unfolding. The transition is occurringreversibly, as evidenced by the perfect overlap between the stretching and relaxingcurves. (b) If the molecule is maintained at a constant force near the transition, it“hops” between the folded and unfolded states. Increasing the force through thetransition, the hairpin ranges from being predominantly folded (13.6 pN, bottomcurve), to being predominantly unfolded (15.2 pN, top). At the midway point, at aforce of 14.5 pN (F1/2), the molecule spends half its time in the folded state and halfin the unfolded state.


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of unfolding was determined, giving �G0 � 113 30 kJ/mol, a value that agreeswell with the predicted free energy value of 147 kJ/mol for unfolding in solution (4).

Effect of Force on the Kinetics of a Reaction

The effect of a mechanical force on the kinetics of a reaction was first describedby Bell in 1978, in the context of cellular adhesion (6). Here we apply the conceptof a tilted free energy surface to describe the force dependence of kinetic rates.For reactions occurring in solution where inertial forces are negligible, the theoryof Kramers (7) gives the rates of transitions between states A and B as

kA3 B ��A�‡

2��/me��G0‡/kBT , kB3 A �

�B�‡

2��/me�(�G0 � �G0‡)/kBT. 12.

The rates depend exponentially on the activation free energies (differencesbetween free energies of the transition state ‡ and states A and B, for the forwardand reverse reactions, respectively). The pre-exponential factors in Equation 12are related to the rates at which the molecule diffuses1 to the transition state fromstates A and B, respectively; they depend on the characteristic frequency �A (�B)of the harmonic potential well at state A (B), which sets the rate at which themolecule “attempts” to overcome the barrier, �‡, which sets the rate of passageover the transition state once it has been reached, and the ratio of the frictioncoefficient experienced by the molecule to its mass, �/m, which is the dampingrate. The characteristic frequencies depend on the curvature of the free energysurface at each state (�A

2 � G�(xA)/m, etc.). Because force “tilts” the free energyalong the mechanical reaction coordinate, if an external force assists the forwardtransition A3B, then the transition state free energy relative to that of state A islowered by an amount F�x‡

A3B, where �x‡A3B � x‡-xA (see Figure 1).

Conversely, the free energy difference between state B and the transition state ‡is increased by F�x‡

B3A � F(�xA3B � �x‡A3B) � F(xB-x‡). As a result, the

forward and reverse rates are modified exponentially by the external force:

13.

Here, we have again assumed that the locations of A, B, and the transition stateare force independent, which as discussed above is not true in general. Positiveforce shifts the positions of minima to longer extensions, while the positions of

1Global protein displacements, such as the movement of a molecular motor along its track(discussed below), are expected to occur in the overdamped limit and are well describedby diffusion. In cases where conformational changes are underdamped, the prefactors inEquation 12 are modified and do not depend on the damping rate (7). In the Eyring model(8), applicable when covalent bonds are made or broken, the prefactor corresponds to asingle quantum mechanical vibrational frequency of the molecule. In all of these models,however, the exponential dependence of the transition rate on the barrier free energy ismaintained.


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maxima are shifted to shorter extensions. Thus, the distance changes, �x‡A3B

and �x‡B3A, that affect the rates are altered with applied force, no matter what

the local curvature of the free energy surface. When states A, B, and ‡ have steepcurvature, these shifts are negligible. For molecular unfolding, this is not thecase, as is discussed in the following section.

Returning to our example of the RNA hairpin, we illustrate what can belearned by studying the force-dependent kinetics of the folding-unfolding reac-tion. Remarkably, by holding the force constant at a value near 15 pN, the hairpinis seen to “hop” between its folded and unfolded states (Figure 2b). Thus, thisexperiment makes it possible to follow the reversible unfolding of a singlemolecule in real time. No intermediates are observed and the reaction can betreated as a cooperative, two-state process. The distributions of dwell times in thefolded state and in the unfolded state give the unfolding and refolding ratecoefficients, respectively, whose force dependence can be determined. As shownin Equation 13, the force dependence of the rate coefficient for unfolding givesthe distance from the folded to the transition state �x‡

f3u along the mechanicalreaction coordinate:

kf3 u(F) � kf3 u0 exp

F�x‡f3 u

kBT. 14.

where kf3u0 is the unfolding rate constant along this pathway at zero force. For

this molecule, the transition state is found to be midway between the folded andunfolded states (�x‡

f3u � 11.9 nm and �x‡u3f � 11.5 nm).

The equilibrium constant for the fºu reaction and its force dependence canalso be determined from the ratio of dwell times of the molecule in the unfoldedand folded states at any given force:

Keq(F) � �u(F)/�f(F). 15.

As shown in Figure 2b, the hopping of the hairpin from the folded to the unfoldedstate depends on the force. By determining the probability of populating thefolded and unfolded states as a function of force, the midpoint of the transitionis found to occur at F1/2 � 14.5 pN. At this force, the molecule spends half itstime folded and half unfolded, and Keq � 1. From Equation 10, a value of 149 16 kJ/mol is determined for the free energy of unfolding, �G0. Keq(F) is foundto depend exponentially on applied force, as predicted by Equation 7 (4). Fromthe force dependence of the equilibrium constant, the distance �xf3u between thefolded and unfolded states is calculated to be 23 4 nm, which agrees well withthe expected length increase upon opening a 22-bp hairpin. By extrapolatingKeq(F) to zero force, �G0 can also be calculated. The value obtained by thismethod, 156 8 kJ/mol, agrees well with that found from the area under theforce-extension curve and from F1/2. In general, because the kinetics of areaction is pathway-dependent, extrapolation of kinetically determined param-


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eters to zero force can give misleading results. We return to this point in thefollowing section.

MECHANICAL UNFOLDING

Introduction

Many biological molecules have a defined mechanical function. For thesemolecules, their resistance to unfolding in response to an applied mechanicalforce—their mechanical stability—is of critical physiological importance. Forexample, titin is the protein responsible for passive elasticity in the skeletal andcardiac muscle sarcomere, where it functions as a molecular spring and ensuresthe return of the sarcomere to its initial dimensions after muscle relaxation(9–11). Fibronectin and tenascin are components of the extracellular matrix,where they extend and contract to facilitate, for example, cell migration andadhesion (12, 13). Regulation of the latter is thought to be controlled by aforce-dependent recognition site in fibronectin (14). These mechanical proteinshave in common a tandem arrangement of -barrel domains, linked by segmentsof unstructured polypeptides. In contrast, spectrin is an -helical protein thatplays a central role in the mechanical properties of erythrocytes, which mustdeform and squeeze through narrow blood vessels during flow (37).

Whereas the examples above demonstrate the importance of a molecule’smechanical stability to its own function, many cellular processes involve theunfolding of macromolecules. For instance, secondary and higher-order nucleicacid structures have to be disrupted to permit translocation by RNA and DNApolymerases, the ribosome, and DNA and RNA helicases. An increasing body ofevidence indicates that these machine-like molecules exert mechanical force ontheir substrates to perform their cellular functions. Similarly, examples of protein“unfoldases” include the import machinery of organelles (15), proteasomes (15,16), and chaperonins (17–18), all of which use chemical energy from ATP toactively unfold (or fold) proteins. Many of these unfolding processes are likelyto be mechanical in nature, although their direct characterization is only nowbecoming experimentally possible.

While direct measurements of forces in vivo during these types of cellularprocesses await future technical developments, much can be learned by studyingwell-defined model systems in vitro and characterizing their responses to force.The development of techniques for manipulating and exerting force on individualmolecules enables us to define, for the first time, the conditions under which amolecule unfolds in response to an applied mechanical force. These early studieshave revealed a broad distribution of mechanical stabilities among macromole-cules, and have found that a molecule’s mechanical stability cannot be predictedfrom its thermodynamic stability. Thus, mechanical stability is a property notdirectly accessible through bulk experiments, and must be determined by direct


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mechanical measurements. Here, we discuss the mechanical unfolding of pro-teins and nucleic acids, and how parameters such as the magnitude, direction, andtime-dependence of the applied force affect the mechanical stability of thesemacromolecules.

Irreversibility in Mechanical Unfolding Experiments

When the mechanical unfolding of a macromolecule occurs at equilibrium, it ispossible to determine directly the free energy, equilibrium constant and kineticsof the reaction and their dependence on force. We discussed in the previoussection how this information can be obtained from reversible, coincident foldingand unfolding curves for the example of the mechanical unfolding of a simpleRNA hairpin. When the extension and relaxation curves do not overlap, folding/unfolding transitions are not occurring reversibly. From the second law ofthermodynamics, the average work done to mechanically unfold the molecule isgreater than the free energy of unfolding:

Wirrev � �G. 16.

Mechanical unfolding under these conditions is less than 100% efficient becausenot all of the mechanical work put in to unfold the molecule is converted to achange in the free energy of the molecule. However, for a two-state system, it hasrecently been demonstrated that it is possible to recover the free energy ofunfolding even when the reaction is not occurring at equilibrium (19–21). Thisresult takes advantage of the ability of single-molecule experiments to providethe distribution of unfolding forces (and hence, work done), rather than just themean value.

More often than not in mechanical unfolding experiments, hysteresis isobserved between the extension and relaxation curves, indicating that themolecule is being extended or relaxed at a rate faster than its rate of equilibration.For the molecule to equilibrate during stretching or relaxation, the total changein force applied to the molecule during its slowest relaxation time � must be lessthan the root-mean-square force fluctuations it would experience at equilibrium,i.e., r� � �Frms, where the loading rate r ' dF/dt (pN/sec) (1). This is therequirement for quasistatic equilibrium during stretching or relaxation. Althoughmost single-molecule mechanical unfolding experiments are performed undernonequilibrium conditions, the observed unfolding force distribution can provideuseful information about the free energy surface, such as the position of thetransition state. The observed unfolding force distribution is peaked, and the mostprobable unfolding force Fu* increases with loading rate as (1)

Fu* �

kBT

�xf3 u‡ ln( r�xf3u

‡

kf3u0 kBT) . 17.

This maximum arises from the convolution of two competing trends: theprobability that a domain remains folded decreases with time (and hence with


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force, since typically F � 1n(r), whereas the probability of unfolding increaseswith force (22). The slope of a plot of Fu* versus 1n(r) yields �xf3u

‡ , whereas theintercept gives the unfolding rate along the mechanical reaction coordinate atzero force kf3u

0 .Why the unfolding force depends on loading rate can be understood by

considering the rate of energy input that drives molecular unfolding. Recallingthat kf3u

0 � A exp(��G‡/kBT) (Equation 12, where here we denote with A theexponential prefactor), we can rewrite Equation 17 as

Fu* �

�G‡

�xf3 u‡ �

kBT

�xf3 u‡ ln(r�xf3 u

‡

AkBT ) . 18.

The second term vanishes whenever

r�xf3 u‡ � AkBT. 19.

The term on the left represents the rate of energy delivery into the system fromthe pulling process; the term on the right represents the rate of energy exchangewith the surrounding thermal bath. Under balanced energy exchange conditions,where these are equal, the unfolding force is equal to the ratio of activationenergy to the distance to the barrier. If r�xf3u

‡ ��AkBT, by contrast, Fu*��G‡/�xf3u

‡ and the process is largely thermally activated. In most unfoldingexperiments, however, we are far from this limit, and r��AkBT/�xf3u

‡ , energy isput into the system faster than it can be dissipated, and Fu*��G‡/�xf3u

‡ .Although pulling at a fixed loading rate is experimentally possible (23), most

unfolding experiments instead have stretched the molecule at a constant speed.Because the stiffness of the molecule depends on the applied force, the loadingrate varies as the molecule is stretched and Equation 17 cannot be used directly(22, 24). Instead, values of kf3u

0 and �xf3u‡ are typically determined with the help

of Monte Carlo simulations, which mimic the stochastic nature of thermallydriven unfolding for a molecule stretched at a constant rate. Values for kf3u

0 areless well determined than those of �xf3u

‡ because the former depend exponen-tially on �xf3u

‡ (25, 26). Representative values for Fu (at a particular pullingspeed), kf3u

0 and �xf3u‡ are listed in Table 1 for various proteins.

The Unfolding Pathway

Because the location of the transition state along the mechanical reactioncoordinate can be determined in these experiments, it should be possible to testfor the presence of intermediates along the reaction pathway. In a two-statesystem, the sum of the distances to the unfolding and refolding transition stateshould equal to the distance along the unfolding reaction coordinate betweennative and denatured states:

�xf3 u‡ � �xu3 f

‡ � �xf3 u. 20.


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TA

BL

E1

Mec

hani

cal

prop

ertie

sof

sele

cted

mol

ecul

es

Mol

ecul

e�

G0

(kca

l/mol

)aF

u(p

N)b

�X

f3u

‡(n

m)

k f3

u0

(s�

1)

Stru

ctur

eat

“bre

akpo

int”

Ref

eren

ce

dsD

NA

1.5–

3(p

erba

sepa

ir)

9–20

c—

—D

NA

base

pair

s(1

10,

111)

P5ab

hair

pin

37.5

4.

814

.5

0.4d

11.9

—R

NA

base

pair

s(4

)

P5ab

c�A

thre

e-he

lixju

nctio

n�

Mg2

�34

.4

4.8

11.4

0.

5d

12—

RN

Aba

sepa

irs

(4)

P5ab

cth

ree-

helix

junc

tion

�M

g2�

36.3

e19

3

(at

3pN

/sw

ithop

tical

twee

zers

)

1.6

0.

12

�10

�4

Hai

rpin

-bul

ge(4

)

127

dom

ain

from

titin

(unf

oldi

ngfr

omna

tive

stat

e)7.

6

100

0.59

1�

10�

6Pa

ralle

l

-she

etsh

ear

(27)

127

dom

ain

from

titin

(unf

oldi

ngfr

omin

term

edia

te)

—20

4

260.

253.

3�

10�

4Pa

ralle

l

-she

etsh

ear

(112

)

128

dom

ain

from

titin

3.0

257

27

0.25

2.8

�10

�5

or1.

9�

10�

6Pa

ralle

l

-she

etsh

ear

(23,

46)

10FN

III

dom

ain

from

fibro

nect

in—

74

200.

382.

0�

10�

2A

ntip

aral

lel

-s

heet

shea

r(1

4,35

)

C2A

dom

ain

—60

——

-s

heet

zipp

er(4

2)

Spec

trin

4.8

0.

5f25

–35

(at

300

nm/s

)1.

7

0.5

3.3

�10

�5

-h

elix

bund

le(3

7)

Ubi

quiti

n(N

-Clin

ked)

6.7g

203

35

(at

400

nm/s

)0.

254

�10

�4

-s

heet

shea

r(3

8)

Ubi

quiti

n(K

48-C

linke

d)6.

7g85

20

(at

300

nm/s

)0.

634

�10

�4

-s

heet

shea

r(3

8)

a From

solu

tion

mea

sure

men

ts,

whe

reav

aila

ble.

Not

eth

atth

erm

odyn

amic

stab

ility

�G

0is

not

am

echa

nica

lpr

oper

ty,

nor

does

itco

rrel

ate

with

mec

hani

cal

stab

ility

,Fu

.bA

llun

fold

ing

forc

esar

equ

oted

for

pulli

ngsp

eeds

inth

eA

FMof

600

nm/s

,un

less

othe

rwis

eno

ted.

c Forc

esob

tain

edva

ryfr

om9

pNfo

rA

Tba

sepa

irs

to20

pNfo

rG

Cba

sepa

irs

and

wer

eth

esa

me

for

optic

altw

eeze

rsan

dA

FMex

peri

men

ts,s

ugge

stin

gth

atun

fold

ing

occu

rsat

equi

libri

um.

dF

uqu

oted

for

thes

ese

cond

ary

stru

ctur

esre

pres

ents

F1/2

,th

efo

rce

atw

hich

the

mol

ecul

ew

aseq

ually

likel

yto

popu

late

the

fold

edan

dun

fold

edst

ates

.T

hese

expe

rim

ents

wer

edo

nein

the

optic

altw

eeze

rsun

der

equi

libri

um,

cons

tant

forc

eco

nditi

ons.

e The

oret

ical

pred

ictio

nfr

omR

efer

ence

4.f Fo

rth

e

16do

mai

nat

25°C

(113

).gR

efer

ence

114.


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See also Figure 1. The P5ab RNA hairpin (discussed in the previous section),which exhibits no folding intermediates, is described reasonably well by thisrelation (�xf3u

‡ ��xu3f‡ � 11.5 � 11.9 nm � 23.4 nm; �xf3u � 23 nm). By

contrast, a finding of �xf3u‡ � �xu3f

‡ � �xf3u may indicate the presence ofintermediates along the reaction pathway. Are there other conditions under whichthe equality of Equation 20 may not hold? We consider the assumptions that gointo determining the distances to the transition state. Equations 13 and 17 assumethat the distance to the unfolding transition state from the folded state, �xf3u

‡ , isindependent of force. Because this distance is in general so short and the well andbarrier are relatively steep, this is a reasonable assumption. By contrast, theseparation between the denatured state and the barrier to refolding, �xu3f

‡ , is notindependent of force, and using the force-dependent refolding kinetics (Equation13) to determine a fixed value of �xu3f

‡ is incorrect (4). This is because thelocation of the free energy minimum for the unfolded state shifts considerablywith force, as described by the worm-like chain equation (2, 3). The distancesobtained for P5ab show reasonable agreement with Equation 20 because theywere determined over a small range of forces: From 13 to 16 pN, the end-to-enddistance of the unfolded RNA chain changes by only 5%, so the distance betweenthe unfolded state minimum and the transition state can be treated as fixed withinthis force range. Over larger ranges of force, however, this distance cannot betreated as constant.

For reactions that possess intermediates (non-two-state), mechanical unfold-ing occurs along a complex free energy surface with multiple energetic barriersand may exhibit different rate-limiting transitions in different ranges of force (27,28). Figure 3 illustrates how force can affect the relative height of barriers alonga three-state unfolding reaction coordinate. The energies and locations of thetransition state barriers and of the intermediate structure strongly influence the

Figure 3 The effect of force on a three-state system, where f, int, and u representthe folded, intermediate and unfolded states of the molecule, respectively. Note howthe rate-limiting barrier changes with applied force.


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unfolding pathway of a molecule under force, since force reduces the free energyto a greater extent at positions further along the mechanical reaction coordinatethan at positions closer to the folded structure. By mechanically unfolding mutantand wild-type I27 domains, and carefully analyzing the unfolding kinetics withinthe constraints of a three-state system, Clarke and coworkers identified distinctunfolding pathways in three different force ranges (27, 29). At high forces, anintermediate state is rapidly attained and unfolding occurs from this state; atlower forces, unfolding occurs directly from the native state; and at forces below43 pN, unfolding follows a distinct “solution” pathway that has a different(lower-energy) transition state than the mechanical transition state. This studydemonstrates the difficulties inherent in extrapolating unfolding rates to zeroforce.

How are the results of mechanical unfolding experiments related to those ofbulk unfolding experiments? Thermodynamic properties such as the free energyof unfolding are state functions and depend only on the initial and final states ofa process; thus, comparisons between single-molecule mechanical studies andbulk biochemical assays should give identical results (after correcting for theentropic contribution of tethered ends). The kinetics of a reaction is, however,pathway dependent. Because single-molecule unfolding experiments impose areaction coordinate different from that of bulk experiments, rates of unfoldingobtained with these two different approaches will generally differ. Figure 4illustrates the effect of force on a two-dimensional free energy surface, where oneof the axes represents the mechanical reaction coordinate and the other anorthogonal “chemical” coordinate. From this simple depiction, it is clear how theunfolding pathway can change with force: an applied force lowers the high-energy barrier located far along the mechanical reaction coordinate. Above agiven force, this barrier becomes lower than the rate-limiting barrier at zero force(which is not located along the mechanical coordinate and is unaffected byforce), creating a more energetically favorable trajectory. Thus, under theinfluence of mechanical force, the unfolding reaction may follow an entirelydifferent trajectory than it does when free in solution (5, 27, 30), as the barrierto unfolding along the mechanical pathway becomes lower in free energy thanthe chemical barrier.

4™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™Figure 4 Force can affect the reaction pathway along a two-dimensional free energysurface. Two paths (A and B) are shown connecting the folded state minimum (lower left)to the unfolded state minimum (upper right) of the free energy contour plot; the preferredpath is indicated by the solid line. The lower panels illustrate the free energy along thesepaths, where blue sections occur along the chemical coordinate and those in red occur alongthe mechanical coordinate. (a) No applied force: the reaction following pathway Aencounters a lower activation barrier than the reaction following pathway B. (b) An appliedforce tilts the free energy surface along the mechanical coordinate, lowering the barrieralong pathway B below the barrier along pathway A.


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Relating Mechanical Stability to Local Molecular Structure

The mechanical unfolding pathways of RNA and proteins are fundamentallydifferent. Complex RNA structures unfold in a hierarchical manner, with stablesecondary domains remaining after tertiary interactions have ruptured. Thishierarchical behavior is rooted in the separability between the energy contribu-tions from the secondary interactions of each individual subdomain and thecontribution of its tertiary contacts (4, 31, 32). In other words, secondary andtertiary interaction energies are independent and additive, and coupling energyterms can be neglected, to a first approximation:

�G � �G2° � �G3° (��Gcoupling) 21.

In agreement with this analysis, single-molecule mechanical unfolding exper-iments have shown that a given secondary RNA domain exhibits the samedistribution of unfolding forces and lengths in isolation as when it is involved intertiary interactions in a larger molecule, meaning that kinetic barriers associatedwith isolated structural subdomains are maintained within the larger molecule(32). Thus, by pulling on progressively larger pieces of the Tetrahymenathermophila L21 ribozyme, it was possible to map completely the unfoldingpathway of the molecule (32). Because tertiary structures have shorter distancesto their transition states than secondary structures, tertiary interactions tend to bebrittle (they break at high forces after small deformation), whereas secondaryinteractions are compliant (they break at low forces and after large deformations)(4). In addition, tertiary interactions equilibrate over a slower timescale thansecondary interactions and their presence is more often than not accompanied byhysteresis in the pulling-relaxation cycle. By pulling the ribozyme many times,it was possible to identify the alternative unfolding pathways of the molecule andtheir relative probability of occurrence (32). The approach described abovesuggests that by characterizing the interaction energies of the various tertiarymotifs (helix-helix, kissing loops, loop-helix, etc.) it should be possible todevelop an Aufbau algorithm to solve the RNA folding problem, i.e., to predictthe tertiary structures of RNA molecules from their sequence using a semiem-pirical approach (31): First the most probable secondary structure is predictedfrom the sequence, and this structure is then used to predict its most probabletertiary fold.

In contrast to RNA, proteins appear to unfold in a highly cooperative manner,and secondary structures are not stable independently of their tertiary context.Mechanical unfolding intermediates have been identified for a few proteins(33–36), in which most intermediate structures involve little disruption of thecore domain of the protein, and instead involve peeling off a single external-strand from a large -barrel or -sandwich structure. In the immunoglobin I27domain, for example, low forces break two hydrogen bonds between outlyingantiparallel -strands, while the core of the protein domain remains folded untilhigher forces are attained (27, 33). The cooperativity of protein unfolding most


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likely arises from the high degree of connectivity involved in the interactions thatmaintain protein structure.

As seen in Table 1, mechanical stability (given by the most probableunfolding force, Fu*) is not correlated with thermodynamic stability (�G), norcan it be predicted from the melting temperature (Tm � �H/�S) (35, 37–39).Instead, as predicted by Equation 17, the mechanical stability of the moleculedepends on the location of the transition state along a specific mechanicalreaction coordinate, the height of the barrier (which contributes to ku

0), and theloading rate. As discussed earlier, force affects most strongly positions furthestfrom the folded, native state: To mechanically eliminate a 10-kBT barrier located0.3 nm from the folded state requires 140 pN of force (when the molecule isstretched under conditions of quasi-static equilibrium), whereas to eliminate abarrier of the same height but located 1.5 nm from the folded state along themechanical reaction coordinate requires only 28 pN under the same conditions.Many mechanical proteins consist of tandem arrays of domains with differenttransition state locations and hence differing mechanical stability. The morecompliant domains (having longer distances to their transition states) stretch inresponse to low applied forces, while the more brittle domains (with shorterdistances to their transition states) maintain their structure to higher appliedforces and only under extreme force conditions would they unfold (35, 36).

The molecular structure provides physical insight into the location of thetransition state barrier along the mechanical reaction coordinate. Proteins thatcontain -sheets tend to unfold at much higher forces and exhibit less compliancethan proteins that are predominantly -helical (see Table 1) (40). Gaub andcoworkers have suggested that this trend can be explained in broad terms by thedifference in forces maintaining tertiary structure (37): -barrel domains are heldtogether by intrasheet hydrogen bonds, which are short-range interactions withshort distances to their transition state, whereas tertiary interactions maintaining-helical bundles are hydrophobic, are more delocalized, and have associatedlonger distances to their transition state. However, due to the local action ofapplied mechanical force, the unfolding force is most strongly influenced notby global domain structure but by the local structure near the mechanical“breakpoint.”

The strongest protein domain examined thus far, the I27 immunoglobindomain from titin, has a -sandwich structure. Structural considerations, molec-ular dynamics simulations, and mutational analysis have been combined todemonstrate that the mechanical breakpoint of this domain at the strongestapplied forces consists of a cluster of six hydrogen bonds between parallel-strands experiencing shear forces (33, 41). Not only is the energy barrier forbreaking many bonds in a concerted fashion much greater than the individualenergy barriers for breaking individual bonds sequentially, but the stiffness of theconnection between the two strands is also greater for shearing six parallelhydrogen bonds than for breaking a single bond (or six bonds in series). Thehigher transition-state energy barrier requires greater forces to lower it; the


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increased stiffness implies steeper walls about the native state free energyminimum, which result in a shorter distance to the transition state for theconcerted versus sequential cleavage of six hydrogen bonds. Similarly shortdistances to the transition state were found for other domains experiencingshearing forces between clusters of hydrogen bonds (Table 1). By contrast,sequential breakage of single hydrogen bonds, occurring when -strands of aprotein are “unzipped” or when an RNA hairpin is mechanically unfolded,requires lower energy, and because the potential energy well is shallower, thetransition state is located at a longer extension. The result is that sequentialcleavage of multiple bonds requires less energy and applied force than concertedcleavage of the same set of bonds. Accordingly, while the structure at themechanical breakpoint is important for mechanical stability, the direction ofapplied force can significantly change the force at which the breakpoint ruptures(38–40, 42, 43).

Extending Single-Molecule Mechanical Properties to theCellular Level

The results of mechanical experiments on individual domains of titin, on singlemolecules of the entire protein, and on the muscle sarcomere illustrate theconvergence, and in some cases complementarity, of information obtained bystudying the mechanical properties of proteins at an increasing level of com-plexity. The similarity of the force-extension curves of single molecules of titinand of the sarcomere (when extrapolated to the single-molecule level) confirmthat single-molecule experiments on titin reproduce the physiological elasticresponse of muscle (36, 44, 45). Muscle elasticity can be described at a still morefundamental level: By determining the mechanical unfolding properties of itsindividual domains, Fernandez and coworkers have reconstructed the force-extension curve of the complete titin molecule (36). They attribute muscleelasticity at low force (� 4 pN) to the compliance of titin’s PEVK and N2Bregions, while the inclusion of the more brittle Ig domains in the tandem arrayprovides titin the “mechanical buffer” necessary to react to potentially damaginghigher forces or to forces applied for long periods of time (36).

The ability to predict, from studies of individual protein domains, themechanical behavior of the entire molecule suggests that, to a first approxima-tion, the domains of titin can be treated as independent structural entities. Thisresult appears to be true for most domains of tandem proteins, in which themechanical unfolding properties of individual domains are independent of theirneighbors. Some domains, however, exhibit different unfolding behavior depend-ing on their context within a tandem array (35, 46, 47). Although the character-ization of individual domains provides valuable information on the mechanicalproperties of the parent protein, the mechanical behavior of the entire moleculemust also be studied to determine the role of interdomain interactions and othersources of higher-order behavior. By repeatedly stretching both individual titinmolecules and the muscle sarcomere, for example, Kellermayer et al. observed


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mechanical fatigue, where the system became more compliant and extended togreater lengths on consecutive pulls (45). This behavior has not been observed instudies of individual domains of titin, and Kellermayer et al. attribute it tononspecific intrachain crosslinks, suggesting that the physiological significanceof fatigue may be to dynamically modulate the mechanical properties of themuscle in response to its recent mechanical history (45). These studies on thesarcomere, titin, and its constituent domains demonstrate how careful experi-ments on systems of varying degrees of complexity can provide insight into thephysiological mechanical behavior of proteins.

MOLECULAR MOTORS

Introduction

To carry out processes as diverse as cell movement, organelle transport, iongradient generation, molecular transport across membranes, protein folding andunfolding, and others, cells possess molecular structures that behave as tinymachine-like devices. These molecular machines must use external energysources to drive directed motion and operate as molecular motors, convertingchemical energy into mechanical work.

The myosin, kinesin, and dynein families of molecular motors, whose best-characterized members are most closely associated with muscle contraction,organelle transport, and ciliary beating, respectively, use ATP hydrolysis as asource of energy to step along a track—actin filaments, in the case of myosin, ormicrotubules, for kinesin and dynein. In replication, transcription, and transla-tion, molecular motors must utilize part of the chemical energy derived from thepolymerization reaction to move along the DNA or RNA in a unidirectionalmanner, generating forces and torques against hydrodynamic drag and/ormechanical roadblocks (48, 49). Many other enzymes that bind to and act onDNA or RNA are molecular motors. Helicases hydrolyze ATP to translocatealong DNA, unwinding it into its complementary strands (50, 51). Type IItopoisomerases exert forces to pass DNA duplexes through double-strand breaksin DNA and to reseal these breaks (52–54). Many protein translocases are alsolikely to operate as molecular motors using ATP hydrolysis to mechanically pullpolypeptide chains across membranes (15). Finally, during the replication cycleof many dsDNA bacteriophages and viruses, a molecular motor must package theDNA of the virus into newly self-assembled capsids against considerableentropic, electrostatic, and elastic forces (55). Motors not only move along lineartracks (microtubules, polymerized actin, DNA, RNA, or polypeptide chain)exerting forces, but can also operate in a rotary fashion, generating torque. F1F0

ATP synthase and the motor at the base of the prokaryotic flagellum utilize theelectrochemical energy of a transmembrane proton gradient to generate torqueand synthesize ATP, and to propel the bacterium, respectively (56, 57).


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Many approaches have been used to study the mechanisms of molecularmotors. Structural studies provide detailed information on the various confor-mations of the motor, but these pictures are static. Biochemical assays of motilityand kinetics provide a more dynamic picture, yet they involve averages overlarge numbers of molecules. Because molecular populations are often heteroge-neous, the ensemble-averaged properties measured in these studies may not berepresentative of individual molecules. The recent development of single-mole-cule techniques has made it possible to probe the individual molecules that makeup the ensemble and, for the first time, allow direct determination of intrinsicmolecular motor properties such as efficiency, stall force, and motor step size. Inthe following section, we discuss in detail the significance of these properties.

Mechanical Properties of Molecular Motors

EFFICIENCY AND COUPLING CONSTANT All molecular motor reactions are exer-gonic (�G � 0), meaning that they occur energetically “downhill.” The freeenergy �G that drives the mechanical motion is derived from chemical sources,for instance, the energy of ATP hydrolysis, the electrochemical energy from atransmembrane ion gradient, or the polymerization energy derived from bondbreaking and forming in DNA, RNA, or polypeptides. As in the case formechanical unfolding, it is possible to define an efficiency—a ratio of outputwork to input energy—of a molecular motor. The thermodynamic efficiency TD

of a motor is defined as the ratio per step between the work done by the motoragainst a conservative external force2 and the free energy associated with thereaction that powers the motor, �G: TD � F�/�G. Here, F is the externalmechanical force and � is the step size of the motor (see below for more on stepsize). The thermodynamic efficiency must be less than unity, since the motorcannot, on average, do more work than the free energy supply �G it is given.Furthermore, since TD increases with the force, it follows that the motor attainsits highest efficiency at the maximum force against which it can work. Asdiscussed below, at this force the motor stalls.

Table 2 lists the thermodynamic efficiencies of a few molecular motors atstall.2 The values vary between 15% to 100%. The observed variation inmolecular motor efficiency may reflect the large uncertainties still associatedwith the determination of the stall force and the step size using methods ofsingle-molecule manipulation (see below). An observed mechanical efficiency�100% suggests that part of the energy of the reaction is either dissipated as heator utilized to perform work along a reaction coordinate orthogonal to thedirection of the applied mechanical load. RNA polymerases, for example, may

2A distinction must be made between the case in which the motor works against aconservative force and a dissipative viscous force. The efficiency of the F1-ATPase (70)was determined from experiments in which the rotary motor operated against the viscousdrag of a long actin filament. Here, a different efficiency from that described above mustbe used: the Stokes efficiency (121).


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have the ability to generate torque to overcome the torsional stress built up in theDNA molecule during transcription. The packaging motor in bacteriophage �29may use part of the energy to twist the DNA and facilitate its arrangement insidethe capsid.

A related concept that quantifies the conversion of chemical energy intomechanical motion is the coupling constant �, defined as the probability that themotor takes a mechanical step per chemical reaction. If one step is taken per catalyticcycle (� � 1), the motor is tight coupled; if less than one step is taken (� � 1), themotor is loose coupled. Still other coupling mechanisms may exist in which manysteps are taken per catalytic cycle (� � 1), so-called one-to-many coupling schemes.

There is no simple correspondence between the coupling � and efficiency .A motor could be tight coupled but have a low efficiency, as appears to be thecase for kinesin (58, 59). In principle, a loose-coupled motor could also have ahigh efficiency. In other words, a motor may step only once per several chemicalreactions, but when it does, utilizes all of the chemical energy available to it.

STALL FORCE The stall force of a molecular motor, Fstall, is the force at whichthe velocity of the motor reduces to zero, and thus it is equal to the maximumforce that the motor itself can generate during its mechanical cycle. Stall forcesvary greatly from motor to motor (Table 2), depending on the motor’s speed ofoperation and step size, which are ultimately dictated by its biological function.For example, during transcription elongation, RNA polymerases must locallyunwind the DNA template, work against torsionally constrained DNA, andpossibly disrupt protein roadblocks such as nucleosomes that impede its trans-

TABLE 2 Mechanochemical properties of a few molecular motors

MotorAverage speed(nm/s)

Average stallforce (pN)

Step size(nm) Efficiencya

F1-ATPaseb 4 rpsh 40 pN�nm 120° 100%

RNA polymerasec 6.8 15–25 0.34 9–22%

DNA polymerased 38 34 0.34 23%

Myosin IIe 8000 3–5 5.3 12–40%

Kinesinf 840 7 8 40–60%

Phage �29g 34 57 0.68 30%

aCalculated from Fstall��/�G where � is the (putative) step size, and �G is the free energy change for the reaction.bReferences 70, 115.cThe step size of RNA polymerase has not been measured directly but is expected to be one base pair (64, 74).dThe step size of DNA polymerase has not been measured directly but is expected to be one base pair (49).eReferences 66, 116–118.fThe stall force for kinesin can vary between 3–7 pN (63, 119, 120).gThe step size of �29 has not been measured directly but is thought to be two base pairs (55, 72).hRevolutions per second (rps).


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location. To perform these tasks, Escherichia coli RNA polymerase generatesforces up to 25 pN (48, 60), sufficient to mechanically unzip dsDNA (15pN;see Table 1). The connector motor at the base of bacteriophage �29’s capsid, on theother hand, must pack the phage DNA in the capsid against the build-up of aninternal pressure that reaches a value of nearly 6 megaPascals (MPa) at the end ofpackaging. To perform this task, the motor is capable of exerting forces ashigh as 60 pN (55).

What factors influence the stall force? The stall force is that at which thetransition of a motor between states at sequential positions along its track occursat equilibrium. In other words, the motor oscillates between these states (A andB) so that the net displacement is zero. Because A and B represent sequentialpositions along a periodic track, the local curvatures of the free energy surface atthese positions are identical, in contrast to those of an unfolding molecule (seeprevious section). As a result, Equation 7 can be used to determine the externalopposing force that stalls the motor (see 61):

Fstall ��G0

��

kBT

�ln

[B]

[A]. 22.

[A] and [B] are the populations of states A and B, respectively, �G0 is thestandard free energy of the reaction, and � is the distance translocated along thetrack: the step size of the motor (see below). Thus, for given values of �G0, [A]and [B], the smaller the step size, the larger the force required to stall the motor.Implicit in Equation 22 is the assumption that the motor is 100% efficient.Indeed, this expression represents the maximum stall force; if the motor utilizesonly a fraction of the reaction free energy, Equation 22 should be scaledaccordingly. Because the magnitude of the stall force depends on the relativepopulations of states A and B, albeit in a weak manner, it is important to specifythe concentrations of products and reactants under which the stall force wasdetermined. For a motor that hydrolyzes ATP, for instance, [B]/[A] in Equation22 is replaced by [ADP][Pi]/[ATP]. A dependence of the stall force on productand reactant concentrations was observed for kinesin (62, 63), but not for E. coliRNA polymerase (48, 64).

When the population of state B is zero (equivalently, when the productconcentration is zero), Equation 22 predicts an infinite stall force. Because thereaction is now irreversible, the motor can never step backward, even under alarge force. The motor will wait until it experiences a thermal fluctuation largeenough to carry it forward. As a result, the velocity becomes exponentially smallat high forces, but will remain positive. In practice, however, there is littleexperimental distinction between an exponentially small and a zero velocity.Thus, many measurements of the stall force likely underestimate the true stallforce (as well as the efficiency), as formally defined in Equation 22. Ultimately,at high enough forces, the motor or its track will deform, rendering it inactive.


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If the reaction is reversible such that the stall force is finite, a motor can inprinciple be run backward, turning products into substrate, by applying suffi-ciently large forces. A possible example of this is ATP synthase in which thedirection of rotation of the F1 motor—by itself an ATPase—is thought to bereversed by the counter-rotating F0 motor in order to drive ATP synthesis (65).The mechanical reversibility of the F1 motor has recently been demonstrated inelegant experiments by Kinosita and coworkers (65a). They mechanicallycounter-rotated the F1 motor and found that the hydrolysis reaction was alsoreversed, synthesizing ATP from ADP and Pi. The work required to reverse amolecular motor must clearly equal or exceed the free energy of the reaction. Ifa motor is 100% efficient, reversal occurs at forces just beyond the stall force asdefined in Equation 22, infinitely slowly, but with 100% efficiency. At largerforces, the process is faster but less efficient.

STEP SIZE The step size, �, of a motor is defined as its net displacement duringone catalytic cycle (see Table 2). Clearly, the step size is best determined fromdirect observations. Several groups have employed single-molecule techniques toobserve individual steps of myosin II (66, 67) and kinesin (68) that compare wellto the known periodicity of their tracks (5.5 nm for actin filaments, 8 nm formicrotubules). Recent experiments (69) suggest that the 8-nm displacement ofkinesin is comprised of two smaller substeps. However, hydrolysis of one ATPmolecule drives the motor the entire 8-nm distance (58, 59).

As discussed above, the distance between two states can vary with force in amanner that depends on the local curvatures of the free energy surface. Because themotor’s track is periodic, we expect sequential positions to have identical free energycurvatures, and thus, we expect the step size to be independent of force. Thisprediction has been confirmed in the case of kinesin (69) and F1-ATPase (70, 71).

In many cases, the step size has not been observed directly. The difficulty liesin the extraordinary spatial resolution required to make such observations.Optical trap techniques do not currently have sensitivity sufficient to resolvesubnanometer displacements. Nevertheless, thermodynamic arguments can bemade to place an upper bound on the step size. Since the thermodynamicefficiency of a motor must be less than 100%, from the free energy of NTPbinding and hydrolysis and the measured stall force a maximum allowable stepsize of 2 basepairs (1 bp � 3.4Å) can be estimated for RNA polymerase (48).For the packaging motor of bacteriophage �29, arguments based on motorefficiency and single-molecule measurements of the maximum stall force(70pN) suggest that the step size must be smaller than 5 base pairs (55). Onebiochemical bulk study that measured the amount of DNA packaged and ATPhydrolyzed suggests that the movement is 2 bps per ATP (72).

Mechanochemistry

The distinguishing feature of a molecular motor is the generation of force toproduce the mechanical motion that accompanies the reaction. How motors


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convert chemical energy into mechanical movement—the mechanochemistry ofthe motor—is discussed in this section. In a sense, force can be considered aproduct of the chemical reaction. Thus, an external force that opposes the motorcan function as an inhibitor of the reaction, and one in the aiding direction canpromote the reaction and act as an activator. As a result, the velocity of the motoroften depends on the external force and does so in a manner dictated by themechanism of motor operation. Force-velocity behavior can thus provide muchinsight into a motor’s mechanochemical conversion. In general, the shape of theforce-velocity relationship will vary depending on the conditions (concentrationof reactants and products of the hydrolysis reaction) under which the motor istested. The rate of a molecular motor will be force dependent if the conditions ofthe experiment are such that movement itself is the rate-determining step.

Figure 5 shows the normalized force-velocity behavior of three molecularmotors—kinesin (73), phage �29 DNA packaging motor (55), and E. coli RNApolymerase (48)—at saturating substrate concentrations. The range in behavioramong motors is striking, and hints at the underlying differences in their

Figure 5 Force-velocity behavior for kinesin (73), RNA polymerase (48), and thebacteriophage �29 packaging motor (55) under saturating conditions. Data werenormalized in order to appear on the same graph. Velocities are scaled to theirmaximum values, and forces are scaled to those at which the velocities are half-maximal, F(v�Vmax/2). The different shapes of the force-velocity curves implydistinct mechanisms.


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respective mechanisms. The velocity of the �29 packaging motor decreaseslinearly with force over practically the entire range of forces, suggesting that therate-limiting step of the reaction, in the conditions in which the data wereobtained, is DNA translocation, even at low forces. Kinesin exhibits similarbehavior, with somewhat less force dependence at low forces than near stall. Thevelocity was practically independent of force when a constant force was appliedassisting the movement of kinesin, suggesting that the rate-limiting step is nottranslocation for assisting forces (73). In marked contrast to those enzymes, thevelocity of RNA polymerase is practically force independent, except near stall,indicating that the movement step is not the rate-limiting step in this case.

The force-velocity relationship of a molecular motor under various concen-trations of substrate and its hydrolysis products provide quantitative informationabout the mechanochemical cycle of the motor. This cycle consists of catalytic stepsthat connect distinct chemical states and mechanical steps that connect differentconformational states. A reaction pathway may typically include a substrate bindingstep, reaction steps, product release steps, and mechanical steps (which may coincidewith chemical steps). Provided the reaction is irreversible (as when the productconcentration is zero, for instance), the rate kt at which the enzyme turns over is givenby an expression with the general form of the Michaelis-Menten Equation:

kt �kcat[S]

KM � [S], 23.

where [S] is the substrate concentration, and where kcat, the maximum rate inunits of moles�sec�1, and KM, the Michaelis constant, are determined by theindividual transition rates connecting the various states of the motor during itsmechanochemical cycle (see below). The velocity of the motor is simply givenby v � ��kt, where � is the step size of the motor, and � is the coupling constant(which we assume is 1 for the following discussion).

Force-velocity curves alone cannot reveal the location of the movement stepin the cycle of the motor. As is illustrated below, it is necessary to determine the forcedependence of the parameters of the Michaelis-Menten Equation, Vmax � ��kcat andKM (61). Consider a general N-step kinetic scheme with two irreversible steps,

M1 � S ¢O¡k1

M2 ¢O¡k2

. . . Mj ¡kj

Mj � 1 ¢O¡k( j�1)

. . . MNO¡kN

M1 24.

where step 1 is the substrate binding step. Because the cycle is irreversible, thevelocity obeys the Michaelis-Menten Equation 23. At saturating substrate levels([S]��KM), the velocity vVmax � ��kcat is independent of substrate. At lowconcentrations ([S]�KM), the velocity is limited by substrate binding:vVmax[S]/KM � ��kcat/KM[S], where kcat/KM is an effective second-order bind-ing rate constant. It can be shown that for the scheme in Equation 24, Vmax

is a function of all rate constants k2, k3,. . . kN except rate constants k1


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associated with the substrate binding step. On the other hand, kcat/KM depends onall of the rate constants that connect enzyme states reversibly to substrate binding(step 1), and on the forward rate for the first irreversible step that follows binding,kj (i.e., k1, k2,. . . kj).

An interesting consequence of this result is that the location of a force-dependent step in the cycle dictates how the parameters Vmax and kcat/KM areaffected by force (see Table 3). There are three possible cases depicted in Figure6 for a simplified reaction cycle, which are discussed below.

THE MOVEMENT STEP COINCIDES WITH BINDING (k�1). This case results in aforce-dependent kcat/KM and a force-independent Vmax. Here, an opposing forceacts like a competitive inhibitor to the substrate, shifting the equilibrium towardthe free enzyme state (M1), reducing the effective binding rate constant kcat/KM.However, the addition of more substrate outcompetes this effect, shifting theequilibrium back to the substrate-bound state (M2). As a result, the velocity atinfinite substrate concentration, Vmax, is unaffected by force. KM increases withopposing force as more substrate is necessary to counteract the effect of force.

™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™3Figure 6 Schematic representation of the free energy surface of a molecular motor alonga generalized reaction coordinate. Sections marked in blue occur along a chemicalcoordinate and are independent of force; those in red occur along the mechanical coordi-nate, and hence depend on force. We assume that the force-dependent step is rate-limitingin each case. For simplicity, we consider the minimal three-state kinetic cycle

M � S ¢O¡k1

M � SO¡k2

M � PO¡k3

M � P,

in which substrate binding is followed by two irreversible steps (depicted by two large freeenergy drops). The effective second-order binding rate kcat/KM is the rate at which the motorreaches the state M�P through the first two steps. Once in that state, the motor is committedto proceeding forward, so that binding cannot be affected by any subsequent transition.Vmax, on the other hand, depends on the second and third transitions, which are independentof substrate concentration. (a) In case 1, the first step is force dependent; kcat/KM decreaseswith force, whereas Vmax is force independent. (b) In case 2, the third transition is forcedependent; Vmax decreases with force, and kcat/KM is force independent. (c) In case 3, thesecond step is force dependent; both kcat/KM and Vmax decrease with force.

TABLE 3 Dependence of parameters from the Michaelis-Menten Equation on opposing force

Case Force-dependent rates Vmax kcat/KM KM

1 k1 – 2 1

2 k(j�1), k(j�2), . . . , kN 2 – 2

3 k2, k3, . . . , kj 2 2 2 or 1


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Recent experiments have shown that the force-velocity behavior of T7 RNApolymerase is consistent with a force-dependent rate-limiting step at subsaturat-ing NTP concentrations (75). This motor exhibits a force-independent velocityVmax at forces below stall, while KM increases with opposing force (75). Thus, T7RNA polymerase appears to follow case 1. The authors have proposed a modelfor translocation in which the movement step occurs in equilibrium with NTPbinding, and hydrolysis and incorporation of NTP serve to rectify translocationin a unidirectional manner.

THE MOVEMENT STEP FOLLOWS THE FIRST IRREVERSIBLE STEP AND PRECEDES THE

NEXT BINDING EVENT (k�(J�1), k�(j�2),. . . kN). In this case, the force dependence isreversed: Vmax depends on force, kcat/KM does not. An opposing force affects theequilibrium between states not connected to substrate binding, like an uncom-petitive inhibitor. As a result, when substrate binding is rate-limiting ([S]�KM),the velocity (v��kcat/KM[S]) does not depend on force. In the other extreme([S]��KM), the velocity (vVmax) is reduced by an opposing force, provided theforce-dependent step (translocation) is rate-limiting. Note that KM must have thesame force-dependence as Vmax in this case to yield a force independent kcat/KM.

As discussed previously, the force-velocity behavior of the �29 packagingmotor at saturating ATP indicates that the rate-limiting step is force dependent.At limiting ATP concentrations, however, it was found that the velocity ispractically independent of force except close to stall (Y.R. Chemla, A. Karunaka-ran, J. Michaelis & C. Bustamante, unpublished data). Measurements of Vmax andKM from the ATP dependence of the velocity at various forces show that bothdecrease with opposing force in the same manner, such that kcat/KM is forceindependent (Y.R. Chemla, A. Karunakaran, J. Michaelis & C. Bustamante,unpublished data). Thus, �29 appears to follow case 2. A putative minimalkinetic scheme for this motor can then be proposed:

M1 � ATP ¢O¡k1

M2O¡k2

M3O¡k3(F)

M4 ¢O¡k4

M5 ¢O¡k5

M1.

The movement step (step 4) occurs after product release.

THE MOVEMENT STEP OCCURS AFTER BINDING IN ANY STEP UP TO AND INCLUDING

THE FIRST IRREVERSIBLE STEP (k�2, k�3,. . . kJ). In this case, force affects theequilibrium between states indirectly connected to binding, so that both Vmax andkcat/KM are force dependent. Here, an opposing force acts essentially as a mixed(noncompetitive) inhibitor. It favors the free enzyme state, reducing the effectivebinding rate kcat/KM (as in case 1), but cannot be outcompeted with substrate, sothat Vmax is still reduced (as in case 2). Note that depending on the relativestrengths of Vmax and kcat/KM, KM may increase or decrease as a function of force.

Two experiments by Block and coworkers (63, 73) have shown that forkinesin Vmax and kcat/KM decrease with opposing force, whereas KM increases,


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consistent with case 3. [By contrast, Nishiyama et al. (69), in an experimentbased on measuring times between kinesin steps, did not observe any forcedependence of KM.] For assisting forces, other transitions in the cycle arerate-limiting, and little force dependence is observed in Vmax and kcat/KM. As aresult of these findings, Block et al. (73) have proposed the following minimalreaction pathway for kinesin:

M1 � ATP ¢O¡k1

M2 ¢O¡k2(F)

M3O¡k3

M4O¡k4(F1)

M5O¡k5(F1)

M1.

Step 2 is the primary movement step and is rate-limiting for opposing forces. Forassisting forces, step 3 is rate-limiting. Steps 4 and 5 are weakly dependent onforces applied perpendicularly to the direction of motion and only affect themaximum velocity Vmax, not the effective binding rate kcat/KM. Furthermore, bystudying the effect of force on ADP binding to kinesin, Uemura et al. (76)showed that assisting force increases the binding affinity of the motor for ADP,whereas an opposing force decreases it. This observation strongly suggests areciprocal coupling between the direction of force (whether assisting or oppos-ing) and the enzymatic activity of the motor. The release of ADP increases thebinding affinity of the kinesin head to the microtubule (77). Uemura et al. (76)postulate that an internal strain may modulate the binding affinity for ADP andhence may control the motor’s processivity.

One of the movement steps of F1-ATPase occurs during binding of ATP (78).Although this appears consistent with case 1, experiments show that Vmax

decreases with torque, an observation more consistent with cases 2 and 3 (70).The reason for this discrepancy is that our treatment of ATP binding as a single-step,second-order process is an oversimplification of the true binding mechanism. Moregenerally, binding may consist of a second-order “docking” process in which thesubstrate comes into loose contact with the binding site, followed by one or morefirst-order “accommodation” steps in which it becomes progressively more tightlybound to the enzyme. Provided these docking steps are reversible, this situation isanalogous to case 3. An illustration of this more complex mechanism is precisely the“binding zipper” in F1-ATPase, in which ATP binding to the catalytic site has beenpostulated to occur via a zippering of hydrogen bonds, generating a constant torque(79). This example illustrates the need to exercise caution when generating mecha-nistic models. Force measurements at various ATP, ADP, and Pi concentrations oftenneed to be supplemented with kinetic measurements and structural data to form acomplete and accurate picture of motor mechanism.

STRAIN IN ENZYME CATALYSIS

Introduction

In the previous section, we have seen that molecular motors have evolved asenergy transducers to convert chemical energy into mechanical work. Binding


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energy represents a ubiquitous but otherwise “silent” form of energy in the cellthat can be considered as one of the main sources of potential energy to performvarious molecular tasks. In this section, we generalize the idea of the energytransducer and discuss how the binding energy between an enzyme and itssubstrate can be utilized to accelerate the rate of chemical reactions.

Chemical catalysis is essential for many critical biological processes toproceed at useful rates under physiological conditions. Thus, a moleculardescription of the mechanisms by which an enzyme achieves high catalyticefficiency is of crucial importance to understand the control and coordination ofthe complex biological reactions in the cell. Studies over the past century havegreatly improved our understanding of the factors that contribute to the catalyticefficiency of enzymes (80, 81). These factors include, among others, the speci-ficity of an enzyme for its substrate, the correct orientation of its reacting groupsat the active site relative to the substrate, and its ability to provide electrostaticshielding of charged intermediates in the reaction. In addition, many mechanismsof rate acceleration have been shown to play an important role in catalysis, suchas general acid-base catalysis, and covalent catalysis. These mechanisms are,however, not sufficient to rationalize the extraordinary rate enhancement pro-vided by enzymes. Although other mechanisms have been considered in the past70 years, most have remained in the realm of hypotheses, due, in part, to thedifficulty of designing experiments to quantify their importance. One suchmechanism is the hypothesis known as strain-induced catalysis. Viewed from thetransition state theory of chemical reactions (82, 83), it is well accepted that tocatalyze biochemical reactions, enzymes must lower the activation barrier for thesubstrate to reach the transition state. Due to the intrinsically different structuresof the substrate and the transition state, it is almost certain that the enzyme activecenter cannot be perfectly complementary to both at the same time. In the 1930s,J.B.S. Haldane suggested that enzymes cannot be efficient catalysts if they arefully complementary to the substrate. Rather, he suggested, enzymes must exertstrain on the substrate upon binding. He wrote, “. . . the key does not fit the lockperfectly but exercises a certain strain on it” (84). This idea was elaborated byPauling in 1946. He pointed out that to accelerate the rate of a reaction, enzymesmust display complementarity to the transition state. He wrote, “The onlyreasonable picture of the catalytic activity of enzymes is that which involves anactive region of the surface of the enzyme which is closely complementary instructure not to the substrate molecule itself in its normal configuration, but ratherto the substrate molecule in a strained configuration, corresponding to the‘activated complex’ for the reaction catalyzed by the enzyme” (85). Combined,these two conjectures are known as the Haldane-Pauling hypothesis.

The Haldane-Pauling hypothesis implies the utilization of binding energy tobring about the acceleration of the rate of the reaction, i.e., strain-inducedcatalysis. To see how the generation of strain could lead to acceleration of thereaction, let us assume that both the enzyme and the substrate are compliant andcan deform upon binding. Let the enzyme’s active site be complementary not to


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the shape of the substrate but to that of the transition state of the reaction. Whenthe substrate first binds, the enzyme is not in a configuration that is catalyticallyproductive. However, because the substrate is not fully complementary to theactive site of the enzyme, and the full complement of binding interactions areonly realized when the substrate attains the transition state, the enzyme exertsstrain on the substrate and this strain favors the attainment of the transition state.It is this gradient of stabilizing binding interactions existing between thesubstrate and its transition state along the reaction coordinate that amounts tothe generation of a force acting on the substrate that ultimately pushes along thereaction. In symbols we can write

�Ebind

�xr

��Ebind

�x�

�x

�xr

, 25.

where Ebind is the binding energy gained through complementarity to the enzymealong the reaction coordinate, xr, and x is the mechanical coordinate throughwhich the substrate is strained. The second factor on the right-hand sideexpresses the decrease of substrate strain as it moves along the reaction coordi-nate toward the transition state, whereas the first represents the effective mechan-ical force exerted by the enzyme on the substrate during this process:

F ��Ebind

�x. 26.

These expressions show that whether the enzyme undergoes a conformationalchange during the process that “drives” the substrate into the transition state orsimply provides a gradient of binding interactions along the reaction coordinatethat preferentially stabilizes the transition state, the result is the same; in bothcases, one can think of the enzyme as exerting a force on the substrate thatcatalyzes the reaction.

In some cases, the enzyme active site may be more compliant than thesubstrate and thus, upon binding, the substrate actually induces strain in theenzyme due to its nonideal fit (86). However, such strain will generate an elasticrestoring force in the enzyme that will be exerted on the substrate and that, inturn, will tend to distort the substrate to bring about the transition state. In otherwords, the strain in the enzyme increases the free energy of the enzyme-substratecomplex along the reaction coordinate and facilitates the crossing of the transi-tion state beyond which such strain is fully relieved.

The Catalytic Advantage of Transition-StateComplementarity

In this model, binding of the substrate to the enzyme induces strain in thecomplex, and complementarity between the enzyme and substrate is fullyrealized only at the transition state. We illustrate below the advantage oftransition-state complementarity in catalysis.


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In Michaelis-Menten kinetics, kcat is a first-order rate constant associated withthe rate of decomposition of the enzyme-substrate complex into enzyme andproducts, whereas kcat/KM is an apparent second-order rate constant for thegeneration of products starting from the free enzyme and the free substrate (80),and is a measure of the catalytic efficiency of the enzyme. In symbols,

E � S¢O¡KM

�GS

ESO¡kcat

�G‡

ES‡ ; E � SO¡kcat/KM

�GT‡

ES‡

where �GS is the binding free energy in the association of the enzyme and thesubstrate, and �G‡ and �GT

‡ are the activation free energies from the EScomplex and from the unbound E � S, respectively (Figure 7).

The importance of transition-state complementarity is illustrated in Figure 7under conditions in which [S]��KM and [S]�KM, and when the active site iscomplementary to the substrate and to the transition state of the reaction (see 80).Here, we assume that the maximum possible free energy of association betweenenzyme and substrate is �GS,max, which arises only when there is full comple-mentarity between enzyme and substrate. If instead, the enzyme is ideallycomplementary to the transition state ES‡, an adverse free energy �Gstrain arisingfrom the strain between the enzyme and substrate contributes to substratebinding: �GS � �GS,max� �Gstrain. (�GS, �GS,max�0 when [S]��KM; and�GS, �GS,max�0 when [S]�KM.) In addition, the free energy of the transitionstate is lowered by strain. Note that the expressions for the rate of the reactiondepend on whether or not the enzyme is saturated with the substrate. However,in both cases, it is catalytically advantageous for the enzyme to be fullycomplementary to the transition state (thus inducing strain in the substrate)because the effect is to raise both KM and kcat. Furthermore, transition-statecomplementarity maximizes the value of kcat/KM, the catalytic efficiency of theenzyme. This conclusion is valid regardless of whether the enzyme is saturatedwith the substrate or not.

Although it is difficult to determine the substrate concentrations in vivo, thecases that have been studied, such as those of the glycolytic pathway (87),indicate that enzymes mostly operate under subsaturation conditions (88), i.e.,

™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™3Figure 7 Illustration of the advantage of transition state complementarity as discussed inthe text. Dashed lines illustrate the free energy when the enzyme binding pocket iscomplementary to the substrate; solid lines illustrate the free energy for the case oftransition state complementarity. Algebraically positive (negative) energies are indicated byarrows pointing upward (downward). (a) [S]��KM. KM, kcat, and kcat/KM are increased bytransition state complementarity, and the rate of catalysis increases with kcat. (b) [S]�KM.KM, kcat, and kcat/KM are increased by transition-state complementarity, and the rate ofcatalysis increases with kcat/KM.


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where substrate concentrations are significantly below their corresponding KMs(see Figure 7b). Thus, maximum rates of activity can be obtained if theseenzymes have evolved to display full complementarity to the transition state suchthat kcat/KM is maximized. Furthermore, a high KM maximizes the concentrationof free enzyme [E]. These two conditions ensure the maximum efficiency undersubsaturating conditions, i.e., v � kcat/KM[E][S] (80).

Experimental Evidence for Strain-Induced Catalysis

It has been difficult to demonstrate unequivocally the importance of strain inenzyme catalysis. This is in part due to the lack of methods to directly monitorthis elusive mechanical property. Even in cases in which enzymes have beencocrystallized with their substrates or with their substrates near or at thetransition state, it is difficult to deduce a priori the energies and forces involvedin the enzyme/substrate complex.

Structural studies of enzymes with and without their substrates show clearconformational changes of the enzyme and/or its substrate. A well-knownexample is hexokinase. Hexokinase catalyzes the transfer of a phosphoryl groupfrom ATP to a substrate, such as glucose or mannose. X-ray crystallographicstudies of yeast hexokinase showed that the binding of glucose induces a largeconformational change in the enzyme (89, 90). Hexokinase comprises two lobesthat form a cleft. Upon binding of the glucose, the cleft between the lobes closes,and the bound glucose becomes surrounded by protein. The substrate-inducedclosing of the cleft in hexokinase provides an excellent example of the induced-fit mechanism (91, 92).

A complementary technique that probes the dynamics of the enzyme-substrateinteraction is fluorescence resonance energy transfer (FRET) (93, 94). A recentFRET study of the enzyme EcoRI, a type II restriction endonuclease, resolvedlarge conformational changes in the N terminus, a region essential for tightbinding of the DNA. The conformational changes revealed by the FRETexperiments are not visible in any of the crystal structures (95).

More experimental evidence for the development of strain during enzymecatalysis has been indirect through the demonstration of two related issues:enzyme complementarity to the transition state, and the utilization of bindingenergy in catalysis. The former argument dates back to the 1940s when Paulingsuggested that complementarity implied that enzymes should bind more stronglyto analogs of the transition state than to the free form of their substrates.Pauling’s ideas have been confirmed experimentally and further supported by thegeneration of catalytic antibodies (81, 96, 97).

Yin et al. (98) recently obtained direct physical evidence for substrate strainin antibody catalysis. These authors raised an antibody to a strained mimic of asubstrate of the enzyme ferrochelatase. This mimic has a pyrrole nitrogen pushed


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out of planarity, a distortion thought to duplicate that involved in the transitionstate of the reaction (99). Not only does the crystal structure of the antibodycomplexed with the unstrained substrate show the same distortion, but theantibody also catalyzes its metallation by Zn2� with a catalytic efficiencycomparable to that of ferrochelatase. Hokenson et al. (100) studied strain in thecarbonyl bond of the substrate of beta-lactamase. Here, a shift in the carbonylstretch frequency of the beta-lactam substrate upon binding indicated that thisbond is stretched in the ES complex; from the frequency shift, the strain in thecarbonyl bond can be calculated to be about 3%. Values of the strain of abouttwice this value have been found in similar systems (101). A final example showshow release of enzyme strain appears to be involved in the catalysis of thereaction by the enzyme-coenzyme complex between aspartate aminotransferaseand pyridoxal-5�-phosphate (PLP). Here, crystallographic studies indicated thatsteric interactions result in the critical hydrogen bond of the protonated Schiffbase between the enzyme and PLP being strained in an out-of-plane conforma-tion (102). Because this bond is broken in all ES intermediates and in thetransition state, the activation energy to attain the transition state from theunbound E and S is decreased by 16 kJ/mol relative to the unstrained situation,thus decreasing the activation energy of kcat/KM whose value is increased103-fold. Note that strain in the enzyme, as opposed to the more traditional strainin the substrate, is used in the catalysis by this enzyme.

The Magnitude of the Enzyme-Substrate Forces

The three-dimensional structure of proteins is maintained by a large number ofweak interactions and thus, there is a limit to the magnitude of forces and strainsthat can develop between the enzyme and its substrate in the course of a reactionwhile maintaining the enzyme’s structural integrity. Moreover, the mechanicalproperties of enzymes, such as their elastic modulus and compressibility, areexpected to be anisotropic and inhomogeneous, i.e., to vary with their locationwithin the enzyme and to depend on the direction along which they are measured.Thus, it is difficult to treat this problem in all generality without considering thespecific interactions between a particular enzyme and its substrate. Nonetheless,general concepts from the basic physics of deformation can give some insightinto the magnitude of the forces and strains that can result from the formation ofthe ES complex.

It is instructive to consider a simple, elastic, and linear model of the generationof strain in the interaction between enzyme and substrate. Here, we follow thetreatment of Gavish (103). Let the complementary surfaces of the enzyme andsubstrate be depicted as in Figure 8. Here L and l are the length dimensions ofthe protein and the substrate before binding interactions are turned on, and x andy are the change in these quantities as a result of these interactions. Also, �E and�S are the spring constants of the enzyme wall and substrate, respectively. Finally,


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FS and Estrain are the force acting on the substrate and the total elastic energy storedin both substrate and enzyme due to the strain. Then it is easy to show that

y ��E

�S � �E

(l � L), FS ��S�E

�S � �E

(l � L), and Estrain �1

2

�S�E

�S � �E

(l � L)2.

27.

Note that when �E �S the distortion y is half of its maximum possible value (l- L). Under these conditions, the strain and stress on the substrate can be written(103)

Strain �n�A(l � L)

2VS

; Stress � �1

n�A(l � L)

2VS

28.

where n is the number of interactions between the substrate and the enzyme, �Ais the area of interaction in the substrate, the isothermal compressibility of theprotein, and VS is the volume of the substrate. Using l - L 2 Å, �A 10 Å2,VS 200 Å3 and 10�6 atm�1, Gavish obtained values of strain 4%,stress 2 � 104 atm, and a stored strain energy of 1 kcal/mol per interaction.These values correspond to a force of 50 pN exerted through each interaction

Figure 8 Pictorial representation of the strain induced between the binding pocketof the enzyme and its substrate. (left) The enzyme and substrate in their unboundconformations with initial lengths L and l, respectively. (right) Distortion is inducedupon binding. The enzyme and substrate distort to achieve a net length change (l-L),where the distortion is shared between the enzyme and the substrate according to theirrelative stiffnesses �E and �S, as discussed in the text.


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on the substrate. For multiple enzyme-substrate interactions, the stored strainenergy becomes considerable, and for four interactions would give rise to acatalytic rate enhancement of 103.

This simple model and calculation show that several interactions working onthe substrate can lead to the development of significant stress and strain in thesubstrate. Indeed, the enzyme and substrate are engaged through multipleinteractions, and it is the sum of all these interactions, each of them weak ifcompared to a covalent bond, that ultimately determines the degree of straininduced in the substrate. Moreover, an enzyme may distort a substrate by pullingin some locations and pushing in others, thus amplifying the mechanical gain (itsmechanical advantage) by using a lever design.

Proteins are likely to be more difficult to compress than to stretch. Asdiscussed above, the maximum force that a protein can support before unfoldingin either case will depend on the rate at which the force is applied. As a result,it is possible that larger forces than those calculated above can develop in theformation of ES complexes if the force is the result of compression of thesubstrate and if it is applied only transiently during the catalytic cycle. Such ascenario may occur when the enzyme possesses a cleft that closes upon substratebinding, as is the case for lysozyme (104–106), for example. In fact, manyauthors have speculated that one reason why enzymes may have evolved to be solarge is to increase their mechanical rigidity (107) and thus to be able to generatelarger stresses and strains on substrates by compression.

Enzymes as Mechanical Devices

Here, we pose the following questions: Are enzymes mechanical devices capableof actively exerting force on their substrates through a coordinated set of motionsakin to the movements of levers and arms of macroscopic machines? Canenzyme movements direct, guide, and mechanically promote the flux andorientation of chemical groups inside the active site as the reaction proceeds(108)? What would be required for these types of machine-like motions to occurin catalysis?

Enzymes undergo concerted conformational changes upon substrate binding,as postulated by the induced-fit hypothesis (91). In this view of enzyme catalysis,the enzyme, fueled by a sequence of successive binding interactions, undergoesa series of motions to bring a substrate into the transition state and to facilitateand push the catalytic process along its reaction coordinate. Indeed, by studyingmolecular motors, we have learned a great deal about how a series of concertedmechanical motions can be coupled to energy sources such as the binding andhydrolysis of fuel molecules. Like molecular motors, enzymes should be able tofunction as mechanical devices undergoing a series of concerted movements byutilizing the potential energy stored in the binding of their substrates or cofactorsor in the release of the product of the reaction.

Imagine the binding of a ligand on the surface of a globular protein. Locally,the target region on the macromolecule is formed with the participation of groups


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from regions both close and far removed from each other in the sequence.Maintaining the architecture of this region of the surface, as that of any other partof the molecule, requires the appropriate balance of forces and interactionsbetween the groups that form the target binding area and between groups locatedfarther away at some radial distance from this area. To gain insight into theability of proteins to behave as mechanical devices capable of “pushing along”the reaction, we need to know how efficiently the strain in one part of themolecule transfers to another because this mechanical communication betweenadjacent regions of the molecule would be needed for the enzyme to carry out aseries of concerted conformational changes. Unfortunately, such information isnot yet available. Some evidence exists that steric strain occurs predominantly inregions involved with function, presumably because of the more stringentprecision needed for ligand binding and catalysis (109). However, much less isknown about changes in strain that result from ligand binding, how far theperturbation brought about by this binding propagates into the macromolecule, orthe contribution of side groups to the strain in the structure. Direct experimentalmeasurements of mechanical forces have provided much insight into the mechan-ical stability of proteins and into the mechanisms by which molecular motors cangenerate force. We anticipate that similar approaches will be used to gain greaterinsight into the ability of enzymes to release stored substrate-binding energy (inthe form of mechanical strain) to catalyze their reactions.

Single-molecule manipulation methods provide the possibility of directlytesting some of the main ideas in strain-induced enzymatic catalysis. Themechanics of the enzyme-substrate interaction can be probed by exerting externalmechanical force on the enzyme, the substrate, or both. For example, by applyingexternal mechanical force to the two lobes of hexokinase, it should be possibleto directly affect either the formation of the ES complex from the free reactantsin solution or, possibly, the attainment of the transition state. If the closure of thecleft is rate-limiting, the external force will significantly affect the rate of thereaction, providing direct evidence for the effect of strain in enzyme catalysis.Moreover, by varying the magnitude of the external force applied to theenzyme-substrate complex, it should be possible to estimate the maximum forcegenerated within the complex.

EPILOGUE

One of the main differences between biological macromolecules and their smallorganic and inorganic counterparts lies in the number, strength, and nature of theinteractions that maintain their three-dimensional structures. The large number ofrelatively weak interactions that stabilize the tertiary structures of macromole-cules also confer on these molecules their unique structural adaptability andflexibility in the cell. Often, this structural pliability is manifested in the form ofa conformational change that ensues from the binding interactions of macromol-


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ecules with other macromolecules or with small ligands. The concept of confor-mational change that has played a central role in traditional biochemical studiesis also essential to the new description proposed in this review: The energyreleased through binding interactions or bond hydrolysis leads to moleculardisplacements and to the generation of forces and mechanical work along acoordinate. The significance of this new description resides in the fact that itmakes it possible to directly associate energies in the form of mechanical workalong a particular coordinate with the corresponding changes in structure. In thenext few years the basic mechanical nature of many more essential biochemicalprocesses will likely be recognized and studied in this fashion. We hope thatthese processes will be amenable to study and characterization by some of thesame concepts and experimental methods described in this review.

The Annual Review of Biochemistry is online at http://biochem.annualreviews.org

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Annual Review of Biochemistry Volume 73, 2004

CONTENTS

THE EXCITEMENT OF DISCOVERY, Alexander Rich 1

MOLECULAR MECHANISMS OF MAMMALIAN DNA REPAIR AND THE DNA DAMAGE CHECKPOINTS, Aziz Sancar, Laura A. Lindsey-Boltz, Keziban Ünsal-Kaçmaz, Stuart Linn 39

CYTOCHROME C -MEDIATED APOPTOSIS, Xuejun Jiang, Xiaodong Wang 87NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY OF HIGH-MOLECULAR-WEIGHT PROTEINS, Vitali Tugarinov, Peter M. Hwang, Lewis E. Kay 107

INCORPORATION OF NONNATURAL AMINO ACIDS INTO PROTEINS, Tamara L. Hendrickson, Valérie de Crécy-Lagard, Paul Schimmel 147

REGULATION OF TELOMERASE BY TELOMERIC PROTEINS, Agata Smogorzewska, Titia de Lange 177CRAWLING TOWARD A UNIFIED MODEL OF CELL MOBILITY: Spatial and Temporal Regulation of Actin Dynamics, Susanne M. Rafelski, Julie A. Theriot 209

ATP-BINDING CASSETTE TRANSPORTERS IN BACTERIA, Amy L. Davidson, Jue Chen 241

STRUCTURAL BASIS OF ION PUMPING BY CA-ATPASE OF THE SARCOPLASMIC RETICULUM, Chikashi Toyoshima, Giuseppe Inesi 269DNA POLYMERASE , THE MITOCHONDRIAL REPLICASE, Laurie S. Kaguni 293

LYSOPHOSPHOLIPID RECEPTORS: Signaling and Biology, Isao Ishii, Nobuyuki Fukushima, Xiaoqin Ye, Jerold Chun 321

PROTEIN MODIFICATION BY SUMO, Erica S. Johnson 355

PYRIDOXAL PHOSPHATE ENZYMES: Mechanistic, Structural, and Evolutionary Considerations, Andrew C. Eliot, Jack F. Kirsch 383

THE SIR2 FAMILY OF PROTEIN DEACETYLASES, Gil Blander, Leonard Guarente 417

INOSITOL 1,4,5-TRISPHOSPHATE RECEPTORS AS SIGNAL INTEGRATORS, Randen L. Patterson, Darren Boehning, Solomon H. Snyder 437

STRUCTURE AND FUNCTION OF TOLC: The Bacterial Exit Duct for Proteins and Drugs, Vassilis Koronakis, Jeyanthy Eswaran, Colin Hughes 467

ROLE OF GLYCOSYLATION IN DEVELOPMENT, Robert S. Haltiwanger, John B. Lowe 491

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STRUCTURAL INSIGHTS INTO THE SIGNAL RECOGNITION PARTICLE, Jennifer A. Doudna, Robert T. Batey 539

PALMITOYLATION OF INTRACELLULAR SIGNALING PROTEINS: Regulation and Function, Jessica E. Smotrys, Maurine E. Linder 559

FLAP ENDONUCLEASE 1: A Central Component of DNA Metabolism, Yuan Liu, Hui-I Kao, Robert A. Bambara 589

EMERGING PRINCIPLES OF CONFORMATION-BASED PRION INHERITANCE, Peter Chien, Jonathan S. Weissman, Angela H. DePace 617

THE MOLECULAR MECHANICS OF EUKARYOTIC TRANSLATION, Lee D. Kapp, Jon R. Lorsch 657

MECHANICAL PROCESSES IN BIOCHEMISTRY, Carlos Bustamante, Yann R. Chemla, Nancy R. Forde, David Izhaky 705

INTERMEDIATE FILAMENTS: Molecular Structure, Assembly Mechanism, and Integration Into Functionally Distinct Intracellular Scaffolds, Harald Herrmann, Ueli Aebi 749

DIRECTED EVOLUTION OF NUCLEIC ACID ENZYMES, Gerald F. Joyce 791

USING PROTEIN FOLDING RATES TO TEST PROTEIN FOLDING THEORIES, Blake Gillespie, Kevin W. Plaxco 837

EUKARYOTIC mRNA DECAPPING, Jeff Coller, Roy Parker 861

NOVEL LIPID MODIFICATIONS OF SECRETED PROTEIN SIGNALS, Randall K. Mann, Philip A. Beachy 891

RETURN OF THE GDI: The GoLoco Motif in Cell Division, Francis S. Willard, Randall J. Kimple, David P. Siderovski 925

OPIOID RECEPTORS, Maria Waldhoer, Selena E. Bartlett, Jennifer L. Whistler 953

STRUCTURAL ASPECTS OF LIGAND BINDING TO AND ELECTRON TRANSFER IN BACTERIAL AND FUNGAL P450S, Olena Pylypenko, Ilme Schlichting 991

ROLES OF N-LINKED GLYCANS IN THE ENDOPLASMIC RETICULUM, Ari Helenius, Markus Aebi 1019

ANALYZING CELLULAR BIOCHEMISTRY IN TERMS OF MOLECULAR NETWORKS, Yu Xia, Haiyuan Yu, Ronald Jansen, Michael Seringhaus, Sarah Baxter, Dov Greenbaum, Hongyu Zhao, Mark Gerstein 1051

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